When you have a counting problem like this, you have to think in terms of how many choices you have. A generic plate is like this:
[tex] x_1x_2x_3x_4x_5x_6,\quad x_1,x_2,x_3 \in \{0,1,\ldots,9\},\quad x_4,x_5,x_6 \in \{a,b,\ldots,z\} [/tex]
so, you have 10 choices for each of the first three places, and 26 choices for each of the last three places.
This leads to a total of
[tex] 10 \cdot 10 \cdot 10 \cdot 26 \cdot 26 \cdot 26 = 10^3 \cdot 26^3 = (10 \cdot 26)^3 = 260^3 = 17576000 [/tex]
possible plates.
